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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

135.0. "Nonlinear periodic recursions" by HARE::STAN () Wed Aug 22 1984 00:04

Shmuel Avital notes that the nonlinear recurrence

		a  + 1
		 n
	a    = --------
	 n+1     a
		  n-1

generates a periodic sequence (of period 5) for any initial a  and a .
							     0      1

He then asks if it is possible to construct nonlinear recursive relations
that generate periodic sequences of any given period length k.

			Reference

Shmuel Avital, Problem 912. Crux Mathematicorum. 2(1984)53.
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135.1HARE::STANWed Aug 22 1984 00:1724
I wrote some programs to hunt for periodic sequences generated by
nonlinear recurrences of the form

		 p a  + q
		    n
	a    =  ----------
	 n+1    r a    + s
		   n-1

and the only ones I found were:

a(n+1) = r / a(n)                generates a sequence of period 2.

a(n+1) = -r^2 / ( a(n) + r )     generates a sequence of period 3.

a(n+1) = -2r^2 / ( a(n) + 2r )   generates a sequence of period 4.

a(n+1) = (r a(n) + r^2) / a(n-1) generates a sequence of period 5.

a(n+1) = -3r^2 / ( a(n) + 3r )   generates a sequence of period 6.

Have I missed any?
Are there any other general or specific forms that generate sequences
of period k with k>6?
135.2HARE::STANWed Aug 22 1984 15:343
Oh, and I forgot to mention

a(n+1) = a(n) / a(n-1)     also generates a sequence of period 6.
135.3TURTLE::GILBERTWed Aug 22 1984 23:0013
Given the form of a recurrence of a    in terms of a  and a   , one approach
				   n+1		    n	   n-1
for finding cycles of length k is to expand a      and a    (in terms of a
					     n+k-1	n+k		  n-1
and a ), and set the expressions equal to a    and a , respectively.
     n					   n-1	    n

This may be a very ambitious undertaking for k > 4 -- for humans, but VAXima
may be able to offer some solutions to the resulting equations.  Note that if
c divides k, and solutions having a cycle length of c is known (even if c=1),
it should be possible to factor these from the expansions for cycle length k.

					- Gilbert
135.4HARE::STANFri Aug 31 1984 23:249
A recursion (for real numbers) generating a sequence of period 9 is

	a     = | a  | - a      .
	 n+1       n      n-1

		Reference
		---------

Morton Brown, Problem 6439, American Mathematical Monthly. 90(1983)569.
135.5HARE::STANWed Sep 26 1984 07:3112
A recursion of period 7 is

		a  = a  = a  = d
		 1    2    3

	a    = |a    - a   |  +  |a    - a |		.
	 n+3	 n+2    n+1	   n+1	  n

			Reference
			---------
David R. Richman, Sums of Absolute Differences. Journal of Recreational
	Mathematics. 17(no. 1)(1984)38-41.
135.6TOOLS::STANTue Jun 04 1985 20:164
The definitive reference showing how to construct non-linear periodic
sequences for any given period length is

R. C. Lyness, "Cycles", Mathematical Gazette. 45(1961)207-209.